Quantum error correcting (QEC) codes protect quantum information against environmental noise. Computational errors caused by the environment change the quantum state within the qubit subspace, whereas quantum erasures correspond to the loss of qubits at known positions. Correcting either type of error involves different correction mechanisms, which makes studying the interplay between erasure and computational errors particularly challenging. In this work, we propose a framework based on the coherent information (CI) of the mixed-state density operator associated to noisy QEC codes, for treating both types of errors together. We show how to rigorously derive different families of statistical mechanics mappings for generic stabilizer QEC codes in the presence of both types of errors. Further, we show that computing the CI for erasure errors only can be done efficiently upon sampling over erasure configurations. We then test our approach on the 2D toric and color codes and compute optimal thresholds for erasure errors only, finding a 50 percent threshold for both codes. This strengthens the notion that both codes share the same optimal thresholds. When considering both computational and erasure errors, the CI of small-size codes yields thresholds in very accurate agreement with established results that have been obtained in the thermodynamic limit. Next, we perform a similar analysis for a low-density parity-check (LDPC) code, the lift-connected surface code. We find a 50 percent threshold under erasure errors alone and, for the first time, derive the exact statistical mechanics mappings in the presence of both computational and erasure errors. We thereby further establish the CI as a practical tool for studying optimal thresholds for code classes beyond topological codes under realistic noise, and as a means for uncovering new relations between QEC codes and statistical physics models.
Quantum error correcting (QEC) codes protect quantum information against environmental noise. Computational errors caused by the environment change the quantum state within the qubit subspace, whereas quantum erasures correspond to the loss of qubits at known positions. Correcting either type of error involves different correction mechanisms, which makes studying the interplay between erasure and computational errors particularly challenging. In this work, we propose a framework based on the coherent information (CI) of the mixed-state density operator associated to noisy QEC codes, for treating both types of errors together. We show how to rigorously derive different families of statistical mechanics mappings for generic stabilizer QEC codes in the presence of both types of errors. Further, we show that computing the CI for erasure errors only can be done efficiently upon sampling over erasure configurations. We then test our approach on the 2D toric and color codes and compute optimal thresholds for erasure errors only, finding a 50 percent threshold for both codes. This strengthens the notion that both codes share the same optimal thresholds. When considering both computational and erasure errors, the CI of small-size codes yields thresholds in very accurate agreement with established results that have been obtained in the thermodynamic limit. Next, we perform a similar analysis for a low-density parity-check (LDPC) code, the lift-connected surface code. We find a 50 percent threshold under erasure errors alone and, for the first time, derive the exact statistical mechanics mappings in the presence of both computational and erasure errors. We thereby further establish the CI as a practical tool for studying optimal thresholds for code classes beyond topological codes under realistic noise, and as a means for uncovering new relations between QEC codes and statistical physics models.
Quantum error correcting (QEC) codes protect quantum information against environmental noise. Computational errors caused by the environment change the quantum state within the qubit subspace, whereas quantum erasures correspond to the loss of qubits at known positions. Correcting either type of error involves different correction mechanisms, which makes studying the interplay between erasure and computational errors particularly challenging. In this work, we propose a framework based on the coherent information (CI) of the mixed-state density operator associated to noisy QEC codes, for treating both types of errors together. We show how to rigorously derive different families of statistical mechanics mappings for generic stabilizer QEC codes in the presence of both types of errors. We observe that the erasure errors enter as a classical average over fully depolarizing channels. Further, we show that computing the CI for erasure errors only can be done efficiently upon sampling over erasure configurations. We then test our approach on the two-dimensional toric and color codes and compute optimal thresholds for erasure errors only, finding a 50 % threshold for both codes. This strengthens the notion that both codes share the same optimal thresholds. When considering both computational and erasure errors, the CI of small-size codes yields thresholds in very accurate agreement with established results that have been obtained in the thermodynamic limit. Next, we perform a similar analysis for a low-density parity-check (LDPC) code—the lift-connected surface code. We find a 50 % threshold under erasure errors alone and, for the first time, derive the exact statistical mechanics mappings in the presence of both computational and erasure errors. We thereby further establish the CI as a practical tool for studying optimal thresholds for code classes beyond topological codes under realistic noise, and as a means for uncovering new relations between QEC codes and statistical physics models.
Quantum error correcting (QEC) codes protect quantum information from decoherence as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, suboptimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered, we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close agreement with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models. Published by the American Physical Society 2024
Quantum error correcting (QEC) codes protect quantum information from decoherence, as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, sub-optimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close accordance with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive practical tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models.
Threshold analyses of quantum error-correcting codes are well established for stochastic error models, in which errors occur randomly with given probabilities. However, errors in actual devices can also be coherent, such as unwanted Z rotations due to imperfect control, which are not captured by stochastic error models. For the surface code, numerical studies have indicated threshold behavior even under coherent errors, but a rigorous proof of threshold existence is lacking. Here we prove that a positive threshold for coherent Z-rotation errors exists for quantum low-density parity-check codes with a bounded number of logical qubits, including the surface code and other topological codes. Specifically, we show that the maximum-likelihood Pauli recovery suppresses the entanglement infidelity exponentially in the code distance up to a prefactor linear in the number of physical qubits whenever the rotation angles lie below a constant that is independent of the code size. The proof combines Fourier analysis to retain the interference among the amplitudes of coherent errors with the cluster expansion of abstract polymer models. Our results expand the theoretical foundation of quantum error correction and offer a statistical-mechanical description of quantum error correction beyond stochastic errors.
Identifying the best families of quantum error correction (QEC) codes for near-term experiments is key to enabling fault-tolerant quantum computing. Ideally, such codes should have low overhead in qubit number, high physical error thresholds, and moderate requirements on qubit connectivity to simplify experiments, while allowing for high logical error suppression. Quantum low-density parity-check (LDPC) codes have been recently shown to provide a path towards QEC with low qubit overhead and small logical error probabilities. Here, we demonstrate that when the dominant errors are erasures, as can be engineered in different quantum computing architectures, quantum LDPC codes additionally provide high thresholds and even stronger logical error suppression in parameter regimes that are accessible to current experiments. Using large-scale QEC numerical simulations, we benchmark the performance of two families of high-rate quantum LDPC codes, namely the so-called Clifford-deformed La-cross codes and the bivariate bicycle codes, under a noise model strongly biased towards erasure errors. Both codes outperform the surface code by offering up to orders of magnitude lower logical error probabilities. Interestingly, we find that this decrease in the logical error probability may not be accompanied by an increase in the code threshold, as different QEC codes benefit differently from large erasure fractions. While here we focus on neutral atom qubits, the results also hold for other quantum platforms, such as trapped ions and superconducting qubits, for which erasure conversion has been demonstrated.
As quantum computers remain susceptible to noise, QEC is essential for preserving logical information during computations. However, the performance of QEC codes breaks down beyond certain noise thresholds, revealing fundamental limits on their ability to protect quantum information. These limits can be characterized using information-theoretic measures such as the coherent information, which quantifies the maximum rate at which logical information can be reliably transmitted through a noisy quantum channel. In this work, we establish a direct connection between the CI and the binary cross-entropy loss used when training neural network decoders. Specifically, we show that the CI constitutes a sharp lower bound on the achievable loss for decoders that track logical operators across noisy channels. To this end, we develop a transformer-based neural network model based on maximum likelihood decoding. We train this network to estimate the CI and evaluate its performance on the surface code under three noise models: code capacity, phenomenological, and circuit-level noise. Our results demonstrate that the network accurately predicts CI and yields threshold estimates that closely match known theoretical limits. When used as a decoder, the network significantly outperforms the minimum weight perfect matching decoder in terms of logical error rate. We also introduce a novel soft post-selection scheme that independently treats uncertainty in both logical operators and relies on confidence-based filtering of the network's output. We prove that such post-selection strategies, based on the MLD cosets, are optimal, and demonstrate their scalability in terms of both logical error rate and abort probability. These findings establish transformer-based architectures as powerful tools for QEC and provide the first numerical evidence supporting the optimality and scalability of MLD-based post-selection.